Physica C 353 (2001) 14±22

www.elsevier.nl/locate/physc

Growth rate limiting mechanisms of YBa2Cu3O7 ®lms manufactured by ex situ processing Vyacheslav F. Solovyov *, Harold J. Wiesmann, Masaki Suenaga Materials and Chemical Sciences Division, Brookhaven National Laboratory, Building 480, Upton, NY 11973, USA Received 1 August 2000; accepted 19 October 2000

Abstract YBa2 Cu3 O7 ®lms were fabricated on SrTiO3 substrates using the BaF2 ex situ process. Precursor ®lms 1, 3 and 5 lm thick were processed in an atmospheric pressure reactor using a gas mixture of oxygen, nitrogen and water vapor. The ®lms were processed at di€erent water vapor pressures and it was observed that the ®lm growth rate was independent of ®lm thickness and proportional to the square root of the water vapor pressure. The dependence of the ®lm growth rate on ®lm area was also investigated for ®lm areas varying from 10 to 160 mm2 . Surprisingly, it was observed that the growth rate was inversely proportional to square root of the area of the ®lm. A theoretical model is developed and applied to the experimental results presented in this paper. The model correctly predicts the inverse square root dependence of the ®lm growth rate on sample area. In addition, it predicts that it will be dicult to process long samples, such as tapes, in simple reactor geometries. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 74.76.Bz; 81.15.Np Keywords: YBCO; Coated conductors; Barium ¯uoride process

1. Introduction YBa2 Cu3 O7 (YBCO) has demonstrated promising superconducting properties for use at liquid N2 temperature with critical current densities Jc > 1 MA cm 2 in self-®eld and an irreversible magnetic ®eld of 7 T. Recent progress in the development of bi-axially textured YBCO on an oxide bu€ered ¯exible metal tape has opened up possibilities of making use of these properties for

* Corresponding author. Tel.: +1-631-344-5437; fax: +1-631344-4071. E-mail address: [email protected] (V.F. Solovyov).

the applications in electric utility and high magnetic ®eld devices [1,2]. These composite conductors are presently made in meter long lengths with critical currents exceeding 200 A (at self-®eld and 75 K) for one centimeter wide tapes [3]. The fabrication processes for these conductors require methods such as ion-beam assisted deposition and pulsed laser deposition which may not be commercially viable for the manufacturing of long lengths of tape. A number of alternative deposition methods are currently under investigation for YBCO and the bu€er layers. One of these is the socalled BaF2 ex situ process for the deposition of YBCO. In this process, a precursor layer, which consists of ®ne grained Y, Cu, and BaF2 , or possibly BaF2 and oxides of Y and Cu, is ®rst

0921-4534/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 0 ) 0 1 6 5 2 - X

V.F. Solovyov et al. / Physica C 353 (2001) 14±22

deposited on a substrate. This layer is then heated in a ¯owing atmosphere which contains an inert gas, a partial pressure of O2 (100 mTorr), and is saturated with H2 O vapor. An attractive aspect of this process is the separation of the deposition of the precursor layer from the heat treatment for the formation of the YBCO. This separation minimizes the requirements for in situ monitoring as well as allows the use of di€erent methods for fabricating the precursor layers, e.g., electron beam evaporation and sol gel processes [4,6,7]. In previous work we have deposited the precursor layer at rates exceeding 10 nm s 1 and rates up to 100 nm s 1 appear feasible. In addition the heat treatment of the precursor layer can be performed in a single step. Because of these favorable attributes, this process for the deposition of YBCO layers has been extensively studied for the development of YBCO thin ®lms [4±7]. More recently, this process is being studied for high current applications of YBCO coated ¯exible conductors with encouraging results at high critical current densities [8±10]. The application of YBCO conductors to electric power or high magnetic ®eld devices requires very high engineering current capacities, >1000 A cm 2 as well as lengths greater than a few hundred meters long. This requirement poses two dicult demands on the processing of practical YBCO conductors, (1) relatively thick layers of YBCO, 5 lm thick or greater, are required and (2) the rate of production or, alternatively, the duration of heat treatment should be within practical limits. This implies that the conversion rate of the precursor to YBCO should be greater than 0.5 nm s 1 as compared to some currently reported growth rates, which are as low as 0.1 nm s 1 . In order to meet these challenging requirements, a thorough understanding of the growth kinetics of the YBCO conversion process is sorely needed. For this reason, we studied the microstructural details of the process of nucleation and growth as well as the rate limiting processes for the growth of YBCO from electron beam evaporated precursor ®lms on SrTiO3 single crystals [8,11,12]. In this article we report our experimental results and theoretical analysis of the processes limiting the growth rate of YBCO layers.

15

2. Experimental procedure and results The precursor ®lms of the stoichiometric composition required for YBCO formation were deposited on polished (1 0 0) SrTiO3 substrates using electron beam evaporation for Y and Cu and thermal evaporation for BaF2 . The heat treatment for the formation of YBa2 Cu3 O7 from the precursor was performed in a quartz tube. The quartz tube was 5 cm in diameter and 1.5 m long. The specimen was suspended at the center of the tube at the end of a quartz specimen holder, which was inserted from the gas exhaust end of the furnace. The specimens were processed at a constant temperature of 735°C under a ¯owing gas mixture at atmospheric pressure. The ¯ow rate was 200 cm3 min 1 (further in the text, unless stated otherwise, standard cubic centimeters are used for the gas ¯ow) and the gas mixture consisted of 100 mTorr of O2 , 25±150 Torr of H2 O with the balance being N2 . The details of the precursor deposition and the heat treatment procedures were given in earlier publication [12]. Also, as previously described, the growth rates of the YBCO ®lms were monitored in situ by measuring the conductance of the ®lms. Since a precursor ®lm becomes an insulator as soon as it is exposed to the processing environment and a YBCO ®lm is semiconducting while it is growing, one can conveniently measure the growth rate by this method. In Fig. 1, the time dependence of ®lm conductance during the heat treatment is shown for ®lms 1, 3, and 5 lm thick processed at water vapor partial pressure p…H2 O† ˆ 25 and 150 Torr. As demonstrated earlier [8], the rate of change of the conductance is approximately constant with the processing period, implying that the growth rate of the YBCO ®lm is approximately constant during the heat treatment. In addition, the growth rates are shown to be independent of ®lm thickness from 1 to 5 lm at both partial pressures of H2 O. These results suggest that the solid-state di€usion of HF or H2 O through the precursor ®lm is not the mechanism limiting the growth of YBCO, but rather a rate-limiting step is independent of the precursor ®lm thickness [11]. If the growth rate is limited by the rate of HF removal from the ®lm surface into the processing

16

V.F. Solovyov et al. / Physica C 353 (2001) 14±22

Fig. 1. Dependence of the YBCO/precursor ®lm conductance on heat treatment time for 1, 3 and 5 lm ®lms processed at 735°C. The results are given for two water vapor pressures, 25 and 150 Torr.

atmosphere, then the growth rate should be in¯uenced by the size of the specimen. This is because the removal rate is controlled by the difference in HF partial pressure at the ®lm surface, ps (HF), and the ambient HF partial pressure, pa (HF), in the chamber far removed from the ®lm surface. This pressure drop is expected to take place over a distance approximately equal to the lateral dimensions of the specimen provided that the specimen size remains small compared to the chamber dimensions. Therefore, larger specimens would experience smaller pressure gradients and the HF removal rate, and by extension the growth rate, would be inversely dependent on the lateral dimension of the specimen. In order to determine the dependence of the growth rate on the ®lm size, we measured the growth rates for a set of approximately square ®lms, where the areas varied from 10 to 160 mm2 . The smallest ®lms, which were about 3 mm on a side, were cut from a single piece of a 3  10 mm2 precursor ®lm. Films with larger areas were constructed by combining several 3  10 mm2 ®lms into a composite sample with an approximately square pattern. In the case of a composite sample, the conductance during the heat treatment period was simultaneously recorded at both the ``upstream'' and ``downstream'' edges of the sample in order to measure the homogeneity of the

growth rate over the length of the specimen. For example, the observed di€erence in the growth rate from edge to edge was less than 10% for a composite specimen of ®ve 3  10 mm2 substrates assembled into an approximate 12  13 mm2 rectangle. Also, we did not detect any in¯uence of the carrier gas ¯ow rates on the growth rate kinetics in the range of 100±500 cm3 min 1 . The carrier gas ¯ow rate experiments were performed using 3  10 mm2 substrates with 1 lm thick YBCO ®lms and were oriented such that the gas ¯ow was directed over the 3 mm width. The results of the measurement of the dependence of the YBCO growth rate, G, on the size of the precursor ®lm area, SF , is shown in Fig. 2. The YBCO ®lms were processed at 735°C with oxygen partial pressure p…O2 † ˆ 100 mTorr, and water vapor partial pressure p…H2 O† ˆ 50 Torr. The solid line represents a ®t to the data by a function G  SF a where a  0:5. Since approximately square specimens were used, the growth rate is seen to be inversely proportional to the length of the edge of the precursor ®lm. In a previous publication we assumed that the reactions in Eq. (1) were only slightly shifted from chemical equilibrium and de®ned an equilibrium constant K relating the partial pressures of H2 O and HF. This is reasonable due to the very slow YBCO growth rates (0.1 nm s 1 ). In addition the

Fig. 2. Dependence of the growth rate, G, on the ®lm area, SF , at 735°C and 50 Torr water partial pressure. The solid line is ®tted to the growth rate and given by G  SF 1=2 .

V.F. Solovyov et al. / Physica C 353 (2001) 14±22

partial pressure of HF, released from the reaction, is very small compared with the water vapor pressure present in the processing atmosphere. Based on these assumptions we concluded that the YBCO growth rate was limited by the removal of HF and correctly predicted the functional dependence of the YBCO growth rate, G, on the partial pressure of H2 O as G  p(H2 O)1=2 [11] which was also observed earlier [8]. The model also presumed that the HF removal was limited by solid state di€usion through the unreacted precursor layer. Solution of the di€usion equation predicted the following time dependence of the YBCO growth rate: G ˆ A=…dp2 2At†1=2 , where A ˆ 2KDHF p…H2 O†1=2 , dp is the starting precursor layer thickness, DHF is the diffusion constant of HF in the precursor and K is the equilibrium constant. The model predicted that the growth rate should be strongly time dependent, contrary to our earlier observations [8,9], see also Fig. 1. We speculated that the constant in time growth rate could possibly be explained in one of two ways. The precursor ®lm might be infused with percolating channels. If the density of channels and the cross sectional area were large enough then the di€usion of HF would be governed by the approximate distance between the channels, dch , and the di€usion would be independent of the precursor ®lm thickness. The second possibility was that the gas-phase di€usion of HF was rate limited at the ®lm surface. In this scenario the growth rate would be dependent on ®lm area in agreement with the 1=2 growth rate dependence presented here, G  SF . This result, combined with our previous result, that the YBCO growth rate is independent of ®lm thickness, is consistent with the idea that the growth rate is limited by the removal rate of HF, which in turn is limited by the rate of HF di€usion from the surface of the precursor ®lm into the processing chamber. Below we present an analysis of the process of HF removal from the sample surface to the processing chamber atmosphere by means of gaseous di€usion and convection. 3. Theoretical analysis In the following section the growth rate, G, of YBCO ®lms is calculated by determining the re-

17

moval rate of HF from the precursor surface into the processing chamber. In order to provide a background for the discussion of the kinetics of YBCO formation we ®rst summarize our previously reported studies on the microstructural development and the chemical reaction path for the formation of a YBCO layer. The growth rate is then determined in the limit of a small specimen (much smaller that the reactor cross-section) and a low ¯ow rate of carrier gas. In this limit a signi®cant simpli®cation can be made in calculating the removal rate for HF. Moreover, the calculated results can be compared directly with our experimental results, since this limit also approximates our present experimental conditions. Since the calculated results for a small specimen have important implications for heat treating a long tape, we will also treat the case of a long tape, but still retain the condition of a low gas ¯ow rate. It will be shown that under such conditions the growth rate becomes impracticably slow for any lengths of the tapes, which are technologically meaningful. 3.1. Chemical reaction path for the YBCO formation Based on our experiments to date [11], we believe that the formation of a YBCO layer from an e-beam evaporated precursor ®lm on SrTiO3 proceeds according to the following steps: (1) In the initial stage of heat treatment, a new oxy-¯uoride compound (Y,Ba)(F,O)2 forms from the precursor, a ®ne grained mixture of Y, Cu, and BaF2 . (2) YBCO grains nucleate at the surface of the (1 0 0) SrTiO3 substrate with the YBCO c-axis perpendicular to the substrate surface. (3) After a continuous 50±100 nm layer of YBCO covers the substrate surface, a liquid layer forms separating the YBCO layer and the unreacted precursor. (4) Further growth of YBCO takes place by the dissolution of the precursor into the liquid and the precipitation of YBCO onto the existing YBCO layer. One of the more important ®ndings in our study of the mechanisms limiting YBCO growth is the fact that there is no detectable ¯uorine in the liquid

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V.F. Solovyov et al. / Physica C 353 (2001) 14±22

layer. However ¯uorine is present in the precursor in the ratio Ba=F ˆ 1=2. This implies that the decomposition of the oxy-¯uoride compound and subsequent release of ¯uorine takes place at the liquid±precursor interface. Based on these results, we hypothesized a chemical reaction path for the formation of YBCO as follows: Y ‡ Cu ‡ BaF2 ‡ O2 ‡ H2 O ! Cu2 O ‡ …Y0:3 Ba0:6 †…O0:15 F0:85 †2 ‡ O2 ‡ H2 O; Cu2 O ‡ …Y0:3 Ba0:6 †…O0:15 F0:85 †2 ‡ O2 ‡ H2 O ! 2HF " ‡ liquid…Y; Ba; Cu; O†; liquid…Y; Ba; Cu; O† ! YBa2 Cu3 O6:1 ;

…1†

where the composition of the oxy-¯uoride is an averaged value from a number of measurements at several locations in four ®lms. 3.2. Small specimen limit To calculate the growth rate, we ®rst need to establish the nature of the gas ¯ow at the sample, i.e. laminar or turbulent and the characteristic residence times. The YBCO processing is performed in a ¯owing gas and the process of the HF removal from the ®lm surface into the ambient chamber atmosphere may be strongly in¯uenced by the type of gas ¯ow. The processing tube is 150 cm long and the sample is mounted on a cantilevered beam inserted along the central axis of the tube. The sample is positioned in the center of the processing tube and is located 2 cm from the leading edge of the cantilevered beam. The type of gas ¯ow in the tube is determined by the Reynolds number, Re, de®ned as Re ˆ 2vxR0 =m, where R0 is the radius of the reactor and v and m are the velocity and the kinematic viscosity of the processing gas, respectively. For ReynoldÕs numbers less than 1200 the ¯ow in the tube is always laminar and for numbers greater than 2200 the ¯ow is always turbulent. The velocity is given by v ˆ Ft =pR20 . Ft is the volume ¯ow rate of a gas at an arbitrary temperature, T(K), where Ft ˆ FT …K†=300 and F is the measured volume ¯ow rate. This correction to

the volume ¯ow rate is required since the ¯ow rate, F, is measured at room temperature 300 K. For the experiments reported here typical values of the gas ¯ow rate are F ˆ 200 cm3 min 1 ˆ 3.33 cm3 s 1 and the calculated temperature corrected velocity is v ˆ 0:57 cm s 1 . Taking the processing tube radius, R0 ˆ 2:5 cm and the kinematic viscosity, m ˆ 1:2 cm2 s 1 at the processing temperature, 735°C, the calculated Reynolds number is Re ˆ 2:4. This is much smaller than the Reynolds number required for the transition to turbulent ¯ow and the gas ¯ow in the reactor is laminar for the present case. At the entrance to the processing tube a boundary layer develops. At a distance from the tube entrance given by the critical entry length, Le ˆ 0:07R0 Re ˆ 0:42 cm the boundary layer in the tube converges and the ¯ow develops a parabolic velocity pro®le. When the gas ¯ow reaches the leading edge of the cantilevered beam a new boundary layer develops. The boundary layer thickness, d, for laminar gas ¯ow is given by d  5x=…Re†1=2 where x is the distance from the leading edge of the beam. The removal of HF from the ®lm surface can be well described by the equation for the di€usion of HF in a steady state system in the laminar ¯ow regime, i.e., [13] Dr2 p…HF†

vrp…HF† ˆ 0

…2†

with the boundary condition of a constant partial pressure p(HF)s over the ®lm surface. D is the di€usivity of HF in the processing gas and the very small perturbation of the water vapor pressure is ignored. The spatial distribution of the gas velocity, v, may be found from the equations of momentum transfer in a laminar ¯ow of gas. A complete solution of Eq. (2) is rather complex and requires a numerical method. However, as described below, the second term can be neglected when the ¯ow of a gas is suciently slow over a small specimen. The second term is often called the convection term, but in the present case, is related to the removal of HF by the gas ¯ow over the surface. In order to justify ignoring the convective term, it is sucient to show that the time, sf , required to transport gas molecules across the specimen surface due to the bulk gas ¯ow is much

V.F. Solovyov et al. / Physica C 353 (2001) 14±22

greater than the time, sd , for the gas molecules to travel the same distance by the di€usion in the bulk gas. For x ˆ 2:0 cm, the distance to the edge of the specimen, d ˆ 6:45 cm which exceeds the distance between the sample and the top of the tube furnace, 2.5 cm, and the gas velocity above the sample can be approximated by v…r† ˆ v…1 r2 =r02 † where r0  1:25 cm. At a distance above the sample equal to the sample width, W ˆ 0:3 cm, the ¯ow velocity is 0.25 cm/s and the residence time, sf ˆ W =v, of the carrier gas over the specimen is 1.2 s. The time required for HF gas molecules to di€use the same distance, sd ˆ W 2 =D, is 0.036 s with D ˆ 2:5 cm2 s 1 where the extrapolated diffusivity for H2 O at atmospheric pressure and 735°C was used for HF. Thus, sf  sd , allowing us to neglect the second term in Eq. (2). The HF concentration pro®le and the mass ¯ux, fHF , may now be obtained from the solution of the Laplace equation, Dr2 p…HF† ˆ 0. Note that in omitting the second term in Eq. (2), we do not necessarily neglect the e€ect of the ¯owing gas in the removal of HF from the surface. The role of the gas ¯ow is to determine the ambient or background HF partial pressure, p(HF)a , at the distance far away from the HF source, i.e., d > W . While the di€usive ¯ow of HF from the ®lm surface into the processing gas is the mechanism for the removal of HF near the surface, the steady state HF removal rate is determined by the difference in the partial pressures of HF at the surface and at the distance away from the surface of the s a specimen, i.e., Dp…HF† ˆ p…HF† p…HF† . As discussed above, reaction Eq. (1) is only slightly shifted from equilibrium. An equilibrium constant, K, can be de®ned relating the partial pressures of H2 O and HF at the interface of the unreacted precursor and the liquid where the decomposition of the oxy-¯uoride takes place. Taking into account the experimental observation that the growth rate is independent of the growth rate on ®lm thickness we assume that the impedance to the ¯ow of these gasses through the precursor is much less than through the ambient gaseous atmosphere. That is, the H2 O and HF partial pressures at the surface of the specimen may be taken approximately equal to the partial pressures at the

19

interface and are related by the following relationship: s

s 1=2

p…HF† =‰p…H2 O† Š

ˆK

…3†

where p(HF)s and p(H2 O)s are the partial pressures of HF and H2 O at the surface, respectively. Also, we will assume that p(H2 O) at the surface and through out the chamber is constant since the amount of H2 O in the carrier gas, 25±150 Torr, is much higher than that of HF in the gas. This assumption will be justi®ed below after we estimate p(HF)s . In order to determine the growth rate, G, the di€usive ¯ux of HF from a specimen was calculated using the tabulated solutions of Laplace equation, Eq. (2) without the convective term. We used the solution for the case of heat transfer from an isothermal object immersed in a thermally conductive media under an isothermal surface [14]. Making the obvious substitutions: temperature ! concentration, heat conductivity ! di€usion coecient, the general solution for the net HF ¯ux, UHF , emanating from the ®lm may be written as follows: UHF ˆ DS Dp…HF†=kT

…4†

where k is BoltzmannÕs constant, T is the processing temperature and S is the tabulated shape factor. The concentrations, n, are expressed as partial pressures using the relation p ˆ nkT . The best approximation for S to our ®lm geometry is one of a thin rectangular plate, with width W and length L, at distance L from the isothermal surface. For this geometry the shape factor found from the table in Ref. [14] is S  2pL= ln…4L=W †. Taking into account that only one surface of the ®lm is emitting HF and W  L for the present case, the shape factor may be approximated as S  3W . In order to relate the total HF ¯ux, UHF , and the YBCO growth rate, G, we note that the area averaged ¯ux, fHF , of HF is given by fHF ˆ UHF =W 2 . From the chemical relationship, Eq. (1), we note that the formation of a single unit cell of YBCO releases four ¯uoride ions, which form four HF molecules. The relationship between the YBCO growth rate and the ¯ux of HF out of the ®lm can be given as:

20

V.F. Solovyov et al. / Physica C 353 (2001) 14±22

G ˆ fHF V =4

…5†

where V is the volume of the YBCO unit cell, 10 22 cm 3 . Using Eqs. (4) and (5) the following expression for the growth rate is obtained: G  3VD‰p…HF†

s

a

p…HF† Š=4kTW :

…6†

The background HF pressure, p(HF)a , may be approximated in terms of the growth rate G from the known ¯ux of HF molecules, which are diluted in the carrier gas having a ¯ow rate Ft : p…HF†a ˆ UHF kT =Ft ˆ fHF W 2 kT =Ft  4GW 2 kT =VFt :

…7†

Combining these expressions, (Eqs. (6) and (7)), we ®nd the growth rate to be: s 1=2

G ˆ 3VDK‰p…H2 O† Š

=4kTW …1 ‡ a†

…8†

where a ˆ 3WD=Ft . The growth rate in Eq. (8) is expressed in terms of the experimental variable p(H2 O)s using the relationship in Eq. (3). Again, this prediction of the dependence of the growth rate on the square root of the partial pressure of H2 O is consistent with the earlier experimental observation [8]. There are two regimes for Eq. (8) depending on the value of a. If this parameter is much less than unity i.e. a  1 then; s 1=2

G  3VDK‰p…H2 O† Š

=4kTW :

…8a†

In the other limiting case, a  1; G  3VFt K‰p…H2 O†s Š1=2 =4kTW 2 :

…8b†

For a  1, the volume ¯ow rate of HF removal from the sample surface by di€usion, WD, is much slower than the removal of HF by the carrier gas ¯ow, Ft , at a point far from the surface. In other words, the carrier gas ¯ow is at a sucient rate and the sample is suciently small so that the amount of HF, which is released from the surface region, is removed from the processing chamber as fast as it di€uses from the surface region. Thus, s a we have a condition, p…HF†  p…HF† and the growth rate does not directly depend on the ¯ow rate of the carrier gas. For the present case, W ˆ 0:3 cm, D ˆ 2:5 cm2 s 1 and Ft …735°C† ˆ 10 cm3 s 1 , we ®nd a  0:23. Eq. (8a) is a valid approximation to the experimental conditions used

in this study. It correctly predicts the experimentally observed dependence of the growth rate G on 1=2 the size of the specimen, i.e., G  SF  W 1 as shown in Fig. (2). It is also consistent with the observation that variations of 100±500 cm3 min 1 in the gas ¯ow rate have no signi®cant e€ect on the growth rate. If the ®lm area increases, such that a  1, the dilution and removal of HF by the carrier gas is insucient to prevent build up of HF in the processing atmosphere. At this point p(HF)a is the factor limiting HF removal. In this regime the growth rate is inversely proportional to the ®lm area and depends linearly on the carrier gas ¯ow as shown in Eq. (8b). Obviously, in the limit of a ®lm with in®nite area, the processing system would reach a state of chemical equilibrium and the ®lm growth would stop. In addition to providing a mechanism for understanding of the growth limiting kinetics for a YBCO ®lm using the BaF2 process, the model allows us to estimate the partial pressure of HF at the ®lm surface using Eq. (8a) and the equilibrium constant in Eq. (3). For example, we obtain s p…HF†  20 mTorr using a growth rate of 0.5 nm s 1 (Fig. 2) for a 10 mm2 ®lm at p…H2 O† ˆ 50 Torr and 735°C. Then, the corresponding equilibrium constant for the reaction K…735°C† is 310 3 (Torr) 1=2 . By obtaining values of K at di€erent temperatures, one can calculate the free energy, the heat of reaction and other thermodynamic parameters of the reaction in Eq. (1). However, more direct and precise measurements of the equilibrium constant may be made by processing samples in an atmosphere with controlled partial pressures of HF. 3.3. Long tape limit In the above analysis, we treated the problem of HF removal from the precursor surface and its e€ect on the YBCO growth rate, in the small specimen limit. For the experimental conditions used in this study Eq. (2) can be simpli®ed by neglecting the second term and this simpli®cation allows us to gain the valuable physical insight regarding the primary growth mechanisms for a YBCO layer using the BaF2 process. However, the

V.F. Solovyov et al. / Physica C 353 (2001) 14±22

processing of technological useful superconductors requires the heat treatment of long tapes. In this case, the second term in Eq. (2) cannot be neglected. In the following section we estimate the inhomogeneity of the growth rate for a long tape processed in a tubular reactor. Consider the case of a long tape being processed in a tube where the tape is positioned on the axis of the tube. The tube axis is designated the x-axis and the carrier gas ¯ows through the tube along the positive x-axis direction. HF is generated along the whole length of the tape. The inhomogeneity of the growth rate arises because the HF concentration increases in the direction of gas ¯ow (positive x-axis direction). This elevates the background partial pressure of HF, p(HF)a , and this, in turn, retards the growth of the downstream portion of the tape. The consequence of this e€ect is reported as the inhomogeneous growth rate of the YBCO for tapes as short as 10 cm [10,15]. We solve a one-dimensional version of Eq. (2), taking into account both convective and di€usive terms. The three-dimensional HF concentration pro®le is replaced by an average concentration, n, at a given x coordinate. This simpli®cation provides an analytical solution, which will help in understanding the physics of the problem at hand. The ambient HF pressure in Eq. (2) is converted to the average molecular HF concentration, n(x), using the relationship p…HF†a ˆ nkT . The equilibrium HF concentration at the tape surface, acs 1=2 cording to Eq. (3), is ns ˆ K‰p…H2 O† Š =kT and the ¯ux of HF molecules produced by a length of tape, dx, is dU ˆ D…ns n†S. We take an approximate value of the shape factor S  dx= ln…W =R†  dx, where W is the tape width and R the reactor tube radius. Changes in the concentration due to the action of the source (generation of HF by the tape) is dn=dt ˆ dU=dV , where dV ˆ Sr dx is the volume of a section of the reactor of length dx and Sr is the cross sectional area of the reactor tube. Combining Eq. (2) with the source term we have: D d2 n=dx2

v dn=dx ‡ D…ns

n†=Sr ˆ 0:

…9†

The x coordinate lies along the tube axis and the carrier gas ¯ows downstream along the positive

21

x-axis direction. D and v are the di€usivity of HF and the carrier gas velocity, as before. The ®rst term is the variation of n along the length of the tape due to the di€usion of HF and the second is related to the gas ¯ow. The last term is the source term for HF, which is determined by the local growth rate of YBCO. Solving Eq. (9) for n(x) gives: n…x† ˆ ns

…ns

n0 †e

x=k

…10†

where k ˆ 2DSr =‰…Ft2 ‡ 4D2 Sr †

1=2

Ft Š

…11†

and n0 is the value of n at x ˆ 0. Substituting the resulting p(HF)a in Eq. (6), we ®nd the dependence of the growth rate along the position of the tape to be: G…x† ˆ G0 exp … x=k†

…12†

where G0 is the growth rate at the front end of a tape and is given by Eq. (8a). The parameter k de®nes the approximate maximum length of tape, which can be processed for a given set of processing conditions and reactor geometry. Depending on the values of the gas ¯ow rate, there are two limiting cases. In the limit of low ¯ow, Ft  F0 , where F0 ˆ 2DSr1=2 , k ˆ Sr1=2 , i.e. k becomes equal to the radius of the tube and does not depend on the ¯ow. In the other limit, Ft  F0 , k ˆ Ft =D. It is instructive to estimate the distance k, along a tape, at which the growth rate will be reduced to 1/2 of the value of the growth rate at the front end of the tape for a typical set of experimental conditions. For example, if we take the values used in this study, i.e., R ˆ 2:5 cm, D ˆ 2:5 cm2 s 1 , and a gas ¯ow rate, Ft …735°C† ˆ 10 cm3 s 1 (at atmospheric pressure), this distance is only 7 cm. Unfortunately our analysis predicts that the utilization of a simple reactor design, i.e. a long tape contained in a long small diameter processing tube, will not work. The growth rate will become impractically small even for rather short lengths of tape. Consider, for example, the processing of a 10 m long tape. Suppose that the maximum acceptable nonuniformity of the growth rate is 25% from one end of the tape to the other. This requires k  40 m and the required volume

22

V.F. Solovyov et al. / Physica C 353 (2001) 14±22

¯ow, Ft is 600 l min 1 . At this ¯ow rate a standard size industrial cylinder of gas would empty in about 12 min. Innovative reactor designs will be required.

Department of Energy, under contract no. AC0298CH10886. References

4. Conclusion We have shown that the rate of removal of HF from the surface of a precursor ®lm into the reaction chamber is the rate-limiting step in the growth of YBCO in the BaF2 ex situ process. The data is well described by a model where the di€usion of HF in the processing atmosphere is the dominant transport mechanism. The fact that the 1=2 growth rate G / SF eliminates both di€usion through the unreacted precursor material and a surface barrier layer as the dominant impedance to HF transport. The model also predicts that processing long tapes in a tubular reactor at atmospheric pressure will require inordinately large gas ¯ows. Special measures will be required to overcome this diculty. Finally, although the experimental results which were discussed here were for electron beam and thermally evaporated precursor ®lms, the analytical results are applicable to the formation of YBCO from BaF2 containing precursor ®lms synthesized by other means such as the sol gel process. Acknowledgements The authors appreciate the valuable discussions with D.O. Welch during the course of this work. This work was performed under auspices of US

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